Let θ be a center of symmetry of a symmetric distribution function F. Consider the following (a) An L-estimator based on a score function J(.). (b) An M-estimator based on a score function ψ(.). (c) A...

Let θ be a center of symmetry of a symmetric distribution function F. Consider the following

(a) An L-estimator based on a score function J(.).

(b) An M-estimator based on a score function ψ(.).

(c) A R-estimator based on a score function ϕ(.).

Which regularity conditions should distribution function F and the respective score functions satisfy in each case so that a first-order asymptotic distributional representation holds? Specify the regularity conditions under which the above estimators are (pairwise) asymptotically first-order equivalent.